Johansen cointegration test
[h,pValue,stat,cValue,mles]
= jcitest(Y)
[h,pValue,stat,cValue,mles]
= jcitest(Y,Name,Value)
Johansen tests assess the null hypothesis H(r) of
cointegration rank less than or equal to r among
the numDims
dimensional time series in Y
against
alternatives H(numDims
) (trace
test)
or H(r+1) (maxeig
test).
The tests also produce maximum likelihood estimates of the parameters
in a vector errorcorrection (VEC) model of the cointegrated series.
[
performs the Johansen
cointegration test on a data matrix h
,pValue
,stat
,cValue
,mles
]
= jcitest(Y
)Y
.
[
performs
the Johansen cointegration test on a data matrix h
,pValue
,stat
,cValue
,mles
]
= jcitest(Y
,Name,Value
)Y
with
additional options specified by one or more Name,Value
pair
arguments.


Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside single quotes (' '
). You can
specify several name and value pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.

Character vector, such as $$\Delta {y}_{t}=C{y}_{t1}+{B}_{1}\Delta {y}_{t1}+\dots +{B}_{q}\Delta {y}_{tq}+DX+{\epsilon}_{t}$$ numDims is
the cointegration rank, then C = AB′,
where A is a numDims byr matrix
of errorcorrection speeds and B is a numDims byr matrix
of basis vectors for the space of cointegrating relations. X contains
any exogenous terms representing deterministic trends in the data.
For maximum likelihood estimation, it is assumed that ε_{t} ~ NID(0,Q),
where Q is the innovations covariance matrix.
Values of
Deterministic terms outside of the cointegrating relations, c_{1} and d_{1}, are identified by projecting constant and linear regression coefficients, respectively, onto the orthogonal complement of A.  

Scalar or vector of nonnegative integers indicating the number q of lagged differences in the VEC(q) model of y_{t}. Lagging and differencing a time series reduce the sample size. Absent any presample values, if y_{t} is defined for t = 1:N, then the lagged series y_{t−k} is defined for t = k + 1:N. Differencing reduces the time base to k+2:N. With q lagged differences, the common time base is q+2:N and the effective sample size is T = N−(q+1). Default: 0  

Character vector, such as
 

Scalar or vector of nominal significance levels for the tests. Values must be between 0.001 and 0.999. Default: 0.05  

Character vector, such as
Character vectors values are expanded to the length of any vector value (the number of tests). Vector values must have equal length. 

Rows of Values of  

Rows of  

Rows of  

Rows of  

Rows of 
If jcitest
fails to reject the null of cointegration rank r = 0, the inference is that the errorcorrection coefficient C
is zero, and the VEC(q) model reduces to a standard
VAR(q) model in first differences. If jcitest
rejects
all cointegration ranks r less than numDims
, the
inference is that C has full rank, and y_{t} is stationary in levels.
The parameters A and B in the reducedrank
VEC(q) model are not uniquely identified, though their product C = AB′ is. jcitest
constructs B
=
V
(:,1:r) using the orthonormal eigenvectors
V
returned by eig
, then renormalizes so that V'*S11*V
= I
, as in [3].
To test linear constraints on the errorcorrection speeds A and the
space of cointegrating relations spanned by B, use jcontest
.
Time series in Y
might be stationary in levels or first differences
(i.e., I(0) or I(1)). Rather than pretesting series for
unit roots (using, e.g., adftest
, pptest
, kpsstest
, or lmctest
), the Johansen procedure formulates the question within the model. An
I(0) series is associated with a standard unit vector in the space of
cointegrating relations, and its presence can be tested using jcontest
.
To convert VEC(q) model parameters in the mles
output to VAR(q+1) model parameters, use vec2var
.
Deterministic cointegration, where cointegrating relations, perhaps
with an intercept, produce stationary series, is the traditional sense of cointegration
introduced by Engle and Granger [1] (see egcitest
). Stochastic cointegration, where cointegrating
relations produce trendstationary series (that is, d0
is nonzero), extends
the definition of cointegration to accommodate a greater variety of economic series.
Unless higherorder trends are actually present in the data, models with fewer restrictions can produce good insample fits, but poor outofsample forecasts.
[1] Engle, R. F. and C. W. J. Granger. "CoIntegration and ErrorCorrection: Representation, Estimation, and Testing." Econometrica. v. 55, 1987, pp. 251–276.
[2] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Johansen, S. LikelihoodBased Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[4] MacKinnon, J. G., A. A. Haug, and L. Michelis. “Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration.” Journal of Applied Econometrics. v. 14, 1999, pp. 563–577.
[5] Turner, P. M. “Testing for Cointegration Using the Johansen Approach: Are We Using the Correct Critical Values?” Journal of Applied Econometrics. v. 24, 2009, pp. 825–831.